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We have learnt the Pythagoras Theorem
a b  c .
2
2
2
Observe that in the proof everything is
perfectly fit to each other - ideal.
What are the differences between
the proof of Pythagoras’ theorem and
mere calculations?
1. Generality – it covers all
possible cases.
2. Reliability
3. Based on known results.
Some `facts’ we used
in the proof.
Area of a square with side a =
Area of a triangle
= (base) x (height) / 2.
Hence two triangles with same
base and height have equal area.
Area of a triangle remains the
same under a rotation of
90 degrees.
a2.
Consider the equation
x  109 y  1
2
2
Claim that there are
no positive integer solutions.
To prove the statement, it is not
enough to try x = 1, 2, 3, …, all
The way to, say, 10,000.
Indeed, the first solution occurs
when x=158070671986249 and
y=15140424455100 !
Hence the `claim’ is refuted.
As any proof is based on certain
known statement(s), which, if you
like, we can consider its/their
proof(s).
It is obviously futile to try to prove
everything.
...P2  P1  Po  P1
This phenomenon is observed more
than 2000 years ago by Greek
philosophers and mathematicians.
Socrates Plato Aristotle
Pythagoras
Euclid
The School of Athens by Raphael.
Vatican Museum.
The Story of Socrates and the
Delphi Oracle
The oracle of Apollo at Delphi
Nickname Sauroctonos – Lizard Slayer!
Archaeological Site of Delphi.
Temple of Apollo at Delphi.
It is proclaimed that Socrates is
the wisest among the mortal.
Socrates says that he is wise only
in this sense: he knows what
he does not know; while as others
claim to know what they don’t.
In Book II of Plato's The Republic, we are told that
Gyges worked as a shepherd for the king.
An earthquake opened up the ground where Gyges'
flock was feeding. Inside this opening he found a
golden ring…..
Gyges’ Ring – reveals the true nature, or
essence, of the being.
Plato (~500BC): The changing thing
is manifestation of something more
essential, something unchanging.
The invariants are known as forms.
Theory of Forms. Allegory of Cave:
Sounds like the movie Matrix?
Related concepts: Atom. Prime numbers.
Elements by Euclid (~400BC)
Inside, you can find a
collection of geometric results.
More significantly, the way these
results are presented is rather unique.
The book starts in chapter one with
definitions.
Definition 1.
A point is that which has no part.
Definition 2.
A line is breadthless length.
Definition 3.
The ends of a line are points.
Definition 4.
A straight line is a line which lies evenly
with the points on itself.
It continues with postulates
or axioms –
statements that are assumed to be
true but no proofs are given. Rather,
They serve as pillars for
proving other statements.
Postulate 1.
To draw a straight line from any point to any
point.
Postulate 2.
To produce a finite straight line continuously in a
straight line.
Postulate 3.
To describe a circle with any center and radius.
Postulate 4.
That all right angles equal one another.
Postulate 5.
That, if a straight line falling on two
straight lines makes the interior angles on
the same side less than two right angles,
the two straight lines, if produced
indefinitely, meet on that side on which are
the angles less than the two right angles.
Euclidean geometry had been
recognized as the paradigm of
Geometry and a good description of our
world for thousands of years.
But it remains that the postulates
are unproven assumptions -- it is
possible to consider other types of
Geometry based on different axioms.
This paradigm shift occurred only in
19th century, when Gauss, Bolya
et al. showed that we may replace the
Parallel Axiom and obtain the
Hyperbolic Geometry or Non-Euclidean
Geometry.
Einstein’s General Theory of
Relativity – space-time could be
non-Euclidean!
Black Hole.